Mathwords logoMathwords

Major Arc

Major Arc

The longer of the two arcs between two points on a circle.

 

Circle with points A (top) and B (right) marked, with the major arc highlighted as the longer path from A to B.

 

 

See also

Minor arc

Key Formula

Major Arc=360°−Minor Arc\text{Major Arc} = 360° - \text{Minor Arc}
Where:
  • Major Arc\text{Major Arc} = The degree measure of the longer arc between two points on the circle
  • Minor Arc\text{Minor Arc} = The degree measure of the shorter arc between the same two points (less than 180°)
  • 360°360° = The total degree measure around any circle

Worked Example

Problem: Points A and B lie on a circle. The minor arc AB measures 110°. Find the measure of the major arc AB.
Step 1: Recall that the two arcs formed by points A and B together make a full circle.
Major Arc+Minor Arc=360°\text{Major Arc} + \text{Minor Arc} = 360°
Step 2: Substitute the known minor arc measure.
Major Arc+110°=360°\text{Major Arc} + 110° = 360°
Step 3: Solve for the major arc by subtracting.
Major Arc=360°−110°=250°\text{Major Arc} = 360° - 110° = 250°
Answer: The major arc AB measures 250°.

Another Example

This example goes beyond finding the degree measure and applies it to compute the physical arc length, which requires the radius.

Problem: A central angle of 70° intercepts a minor arc on a circle with radius 12 cm. Find the arc length of the major arc.
Step 1: Find the degree measure of the major arc.
Major Arc=360°−70°=290°\text{Major Arc} = 360° - 70° = 290°
Step 2: Use the arc length formula. Arc length equals the fraction of the full circumference corresponding to the arc's central angle.
L=θ360°×2πrL = \frac{\theta}{360°} \times 2\pi r
Step 3: Substitute the major arc angle and the radius.
L=290°360°×2π(12)L = \frac{290°}{360°} \times 2\pi(12)
Step 4: Simplify the fraction and compute.
L=2936×24π=696π36=58π3≈60.74 cmL = \frac{29}{36} \times 24\pi = \frac{696\pi}{36} = \frac{58\pi}{3} \approx 60.74 \text{ cm}
Answer: The arc length of the major arc is 58π3≈60.74\frac{58\pi}{3} \approx 60.74 cm.

Frequently Asked Questions

What is the difference between a major arc and a minor arc?
A minor arc is the shorter arc between two points on a circle and measures less than 180°. A major arc is the longer arc between the same two points and measures more than 180°. Together, they always add up to 360°.
How do you name a major arc?
A major arc is named using three letters: the two endpoints and a third point that lies on the arc between them. For example, if points A and B are the endpoints and point C lies on the longer arc, you write arc ACB. This three-letter naming distinguishes it from the minor arc, which is simply written as arc AB.
Can a major arc equal exactly 180°?
No. When two points divide a circle into two arcs of exactly 180° each, both arcs are called semicircles. A major arc must be strictly greater than 180°, and a minor arc must be strictly less than 180°.

Major Arc vs. Minor Arc

Major ArcMinor Arc
DefinitionThe longer arc between two points on a circleThe shorter arc between two points on a circle
Degree measureGreater than 180° and less than 360°Greater than 0° and less than 180°
Formula360° − minor arcEqual to its central angle
Naming conventionThree letters (e.g., arc ACB)Two letters (e.g., arc AB)
RelationshipMajor arc + minor arc = 360°Minor arc + major arc = 360°

Why It Matters

Major arcs appear throughout geometry courses when you work with central angles, inscribed angles, and arc length problems. In real-world applications, they describe the longer path along any curved structure—think of the larger sweep of a clock hand or the longer route around a circular track. Understanding major arcs is also essential for computing areas of major sectors and solving problems involving tangent and secant lines.

Common Mistakes

Mistake: Using only two letters to name a major arc (e.g., writing arc AB when you mean the longer arc).
Correction: Always use three letters to name a major arc, including a point on the arc between the endpoints (e.g., arc ACB). Two letters alone default to the minor arc.
Mistake: Confusing the central angle of a minor arc with the measure of the major arc.
Correction: A central angle gives you the minor arc's measure directly. To get the major arc, you must subtract that angle from 360°. For example, a 100° central angle means the major arc is 260°, not 100°.

Related Terms

  • Minor Arc — The shorter arc; complements major arc to 360°
  • Arc of a Circle — General term for any portion of a circle
  • Circle — The shape on which arcs are defined
  • Point — Endpoints that divide the circle into arcs
  • Central Angle — Angle at center whose measure equals the minor arc
  • Semicircle — Arc of exactly 180°; boundary between major and minor
  • Sector — Region bounded by two radii and an arc
  • Arc Length — Physical length of an arc computed from its angle and radius